Here's what the dev wrote to me about running both:
They work completely independently.
Personally, I don't use the Apple one as I find the scheduling very rigid and hard to use.
It is also not as accurate as AutoSleep, doesn't track naps, and has a major design flaw in how it tracks time in bed, so you often see that time in bed is less than time asleep. Really weird.
But you can use both quite happily and mix and match which parts you want to use.
The only thing to consider is which app you want to use in the Apple Health app Summary views (if you ever use that).
If you select Sleep in Health on the Browse tab,
Then scroll all the way to the bottom.
Then select Data Sources & Access.
Then Edit in the top right.
You can now choose which app is given precedence for sleep tracking in the Health app.
Of course AutoSleep uses its own tracking so that setting only affects Apple Health.